Archives For Quant

gmat-math-strategies-01A while back, we talked about the 4 math strategies that everyone needs to master. Today, I’ve got some additional practice for you with regard to one of those strategies: Testing Cases.

Try this GMATPrep® problem:

* ” If xy + z = x(y + z), which of the following must be true?

“(A) x = 0 and z = 0

“(B) x = 1 and y = 1

“(C) y = 1 and z = 0

“(D) x = 1 or y = 0

“(E) x = 1 or z = 0

How did it go?

This question is called a “theory” question: there are just variables, no real numbers, and the answer depends on some characteristic of a category of numbers, not a specific number or set of numbers. Problem solving theory questions also usually ask what must or could be true (or what must not be true). When we have these kinds of questions, we can use theory to solve—but that can get very confusing very quickly. Testing real numbers to “prove” the theory to yourself will make the work easier.

The question stem contains a given equation:

xy + z = x(y + z)

Whenever the problem gives you a complicated equation, make your life easier: try to simplify the equation before you do any more work.

xy + z = x(y + z)

xy + z = xy + xz

z =  xz

Very interesting! The y term subtracts completely out of the equation. What is the significance of that piece of info?

Nothing absolutely has to be true about the variable y. Glance at your answers. You can cross off (B), (C), and (D) right now!

Next, notice something. I stopped at z = xz. I didn’t divide both sides by z. Why?

In general, never divide by a variable unless you know that the variable does not equal zero. Dividing by zero is an “illegal” move in algebra—and it will cause you to lose a possible solution to the equation, increasing your chances of answering the problem incorrectly.

The best way to finish off this problem is to test possible cases. Notice a couple of things about the answers. First, they give you very specific possibilities to test; you don’t even have to come up with your own numbers to try. Second, answer (A) says that both pieces must be true (“and”) while answer (E) says “or.” Keep that in mind while working through the rest of the problem.

z =  xz

Let’s see. z = 0 would make this equation true, so that is one possibility. This shows up in both remaining answers.

If x = 0, then the right-hand side would become 0. In that case, z would also have to be 0 in order for the equation to be true. That matches answer (A).

If x = 1, then it doesn’t matter what z is; the equation will still be true. That matches answer (E).

Wait a second—what’s going on? Both answers can’t be correct.

Be careful about how you test cases. The question asks what MUST be true. Go back to the starting point that worked for both answers: z = 0.

It’s true that, for example, 0 = (3)(0).

Does z always have to equal 0? Can you come up with a case where z does not equal 0 but the equation is still true?

Try 2 = (1)(2). In this case, z = 2 and x = 1, and the equation is true. Here’s the key to the “and” vs. “or” language. If z = 0, then the equation is always 0 = 0, but if not, then x must be 1; in that case, the equation is z = z. In other words, either x = 1 OR z = 0.

The correct answer is (E).

The above reasoning also proves why answer (A) could be true but doesn’t always have to be true. If both variables are 0, then the equation works, but other combinations are also possible, such as z = 2 and x = 1.

Key Takeaways: Test Cases on Theory Problems

(1) If you didn’t simplify the original equation, and so didn’t know that y didn’t matter, then you still could’ve tested real numbers to narrow down the answers, but it would’ve taken longer. Whenever possible, simplify the given information to make your work easier.

(2) Must Be True problems are usually theory problems. Test some real numbers to help yourself understand the theory and knock out answers. Where possible, use the answer choices to help you decide what to test.

(3) Be careful about how you test those cases! On a must be true question, some or all of the wrong answers could be true some of the time; you’ll need to figure out how to test the cases in such a way that you figure out what must be true all the time, not just what could be true.

 

* GMATPrep® questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.

c-trapHave you heard of the C-Trap? I’m not going to tell you what it is yet. Try this problem from GMATPrep® first and see whether you can avoid it

* “In a certain year, the difference between Mary’s and Jim’s annual salaries was twice the difference between Mary’s and Kate’s annual salaries. If Mary’s annual salary was the highest of the 3 people, what was the average (arithmetic mean) annual salary of the 3 people that year?

“(1) Jim’s annual salary was $30,000 that year.

“(2) Kate’s annual salary was $40,000 that year.”

I’m going to do something I normally never do at this point in an article: I’m going to tell you the correct answer. I’m not going to type the letter, though, so that your eye won’t inadvertently catch it while you’re still working on the problem. The correct answer is the second of the five data sufficiency answer choices.

How did you do? Did you pick that one? Or did you pick the trap answer, the third one?

Here’s where the C-Trap gets its name: on some questions, using the two statements together will be sufficient to answer the question. The trap is that using just one statement alone will also get you there—so you can’t pick answer (C), which says that neither statement alone works.

In the trickiest C-Traps, the two statements look almost the same (as they do in this problem), and the first one doesn’t work. You’re predisposed, then, to assume that the second statement, which seemingly supplies the “same” kind of information, also won’t work. Therefore, you don’t vet the second statement thoroughly enough before dismissing it—and you’ve just fallen into the trap.

How can you dig yourself out? First of all, just because two statements look similar, don’t assume that they either both work or both don’t. The test writers are really good at setting traps, so assume nothing.
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data-sufficiencySome Data Sufficiency questions present you with scenarios: stories that could play out in various complicated ways, depending on the statements. How do you get through these with a minimum of time and fuss?

Try the below problem. (Copyright: me! I was inspired by an OG problem; I’ll tell you which one at the end.)

* “During a week-long sale at a car dealership, the most number of cars sold on  any one day was 12. If at least 2 cars were sold each day, was the average daily number of cars sold during that week more than 6?

“(1) During that week, the second smallest number of cars sold on any one day was 4.

“(2) During that week, the median number of cars sold was 10.”

First, do you see why I described this as a “scenario” problem? All these different days… and some number of cars sold each day… and then they (I!) toss in average and median… and to top it all off, the problem asks for a range (more than 6). Sigh.

Okay, what do we do with this thing?

Because it’s Data Sufficiency, start by establishing the givens. Because it’s a scenario, Draw It Out.

Let’s see. The “highest” day was 12, but it doesn’t say which day of the week that was. So how can you draw this out?

Neither statement provides information about a specific day of the week, either. Rather, they provide information about the least number of sales and the median number of sales.

The use of median is interesting. How do you normally organize numbers when you’re dealing with median?

Bingo! Try organizing the number of sales from smallest to largest. Draw out 7 slots (one for each day) and add the information given in the question stem:

Screen Shot 2014-04-10 at 12.37.53 PM

Now, what about that question? It asks not for the average, but whether the average number of daily sales for the week is more than 6. Does that give you any ideas for an approach to take?

Because it’s a yes/no question, you want to try to “prove” both yes and no for each statement. If you can show that a statement will give you both a yes and a no, then you know that statement is not sufficient. Try this out with statement 1

(1) During that week, the least number of cars sold on any one day was 4.

Draw out a version of the scenario that includes statement (1):

Screen Shot 2014-04-10 at 12.38.22 PM

Can you find a way to make the average less than 6? Keep the first day at 2 and make the other days as small as possible:

Screen Shot 2014-04-10 at 12.38.58 PM

The sum of the numbers is 34. The average is 34 / 7 = a little smaller than 5.

Can you also make the average greater than 6? Try making all the numbers as big as possible:

Screen Shot 2014-04-10 at 12.39.24 PM

(Note: if you’re not sure whether the smallest day could be 4—the wording is a little weird—err on the cautious side and make it 3.)

You may be able to eyeball that and tell it will be greater than 6. If not, calculate: the sum is 67, so the average is just under 10.

Statement (1) is not sufficient because the average might be greater than or less than 6. Cross off answers (A) and (D).

Now, move to statement (2):

(2) During that week, the median number of cars sold was 10.

Again, draw out the scenario (using only the second statement this time!).

Screen Shot 2014-04-10 at 12.39.59 PM

Can you make the average less than 6? Test the smallest numbers you can. The three lowest days could each be 2. Then, the next three days could each be 10.

Screen Shot 2014-04-10 at 12.40.21 PM

The sum is 6 + 30 + 12 = 48. The average is 48 / 7 = just under 7, but bigger than 6. The numbers cannot be made any smaller—you have to have a minimum of 2 a day. Once you hit the median of 10 in the middle slot, you have to have something greater than or equal to the median for the remaining slots to the right.

The smallest possible average is still bigger than 6, so this statement is sufficient to answer the question. The correct answer is (B).

Oh, and the OG question is DS #121 from OG13. If you think you’ve got the concept, test yourself on the OG problem.

 

Key Takeaway: Draw Out Scenarios

(1) Sometimes, these scenarios are so elaborate that people are paralyzed. Pretend your boss just asked you to figure this out. What would you do? You’d just start drawing out possibilities till you figured it out.

(2) On Yes/No DS questions, try to get a Yes answer and a No answer. As soon as you do that, you can label the statement Not Sufficient and move on.

(3) After a while, you might have to go back to your boss and say, “Sorry, I can’t figure this out.” (Translation: you might have to give up and guess.) There isn’t a fantastic way to guess on this one, though I probably wouldn’t guess (E). The statements don’t look obviously helpful at first glance… which means probably at least one of them is!

 

gmat-quant-strategySo you’ve been told over and over that guessing is an important part of the GMAT. But knowing you’re supposed to guess and knowing when you’re supposed to guess are two very different things. Here are a few guidelines for how to decide when to guess.

But first, know that there are two kinds of guesses: random guesses and educated guesses. Both have their place on the GMAT. Random guesses are best for the questions that are so tough, that you don’t even know where to get started. Educated guesses, on the other hand, are useful when you’ve made at least some progress, but aren’t going to get all the way to an answer in time.

Here are a few different scenarios that should end in a guess.

Scenario 1: I’ve read the question twice, and I have no idea what it’s asking.

This one is pretty straightforward. Don’t worry about whether the question is objectively easy or difficult. If it’s too hard for you, it’s not worth doing. In fact, it’s so not worth doing that it’s not even worth your time narrowing down answer choices to make an educated guess. In fact, if it’s that difficult, it may even be better for you to get it wrong!

To make the most of your random guesses, you should use the same answer choice every time. The difference is slight, but it does up your odds of getting some of these random guess right.

Scenario 2: I had a plan, but I hit a wall.

Often, when this happens, you haven’t yet spent 2 minutes on the problem. So why guess? Maybe now you have a better plan for how to get to the answer. I know this is hard to hear, but don’t do it! To stay on pace for the entire section, you have to stay disciplined and that means that you only have one chance to get each question right.

The good news is that no 1 question you get wrong will kill your score. But, 1 question can really hurt your score if you spend too long on it! Once you realize that your plan didn’t work, it’s time to make an educated guess. You’ve already spent more than a minute on this question (hopefully not more than 2!), and you probably have some sense of which answers are more likely to be right. Take another 15 seconds (no more!) and make your best educated guess.

Scenario 3: I got an answer, but it doesn’t match any of the answer choices.

This is another painful one, but it’s an almost identical situation to Scenario 2. It means you either made a calculation error somewhere along the way, or you set the problem up incorrectly to begin with. In an untimed setting, both of these problems would have the same solution: go back over your work and find the mistake. On the GMAT, however, that process is too time-consuming. Plus, even once you find your mistake, you still have to redo all the work!

Once again, though it might hurt, it’s still in your best interest to let the question go. If you can narrow down the answer choices, great (though don’t spend longer than 15 or 20 seconds doing so). If not, don’t worry about it. Just make a random guess and vow to be more careful on the next one (and all the rest after that!).

Scenario 4: I checked my pacing chart and I’m more than 2 minutes behind.

Pacing problems are best dealt with early. If you’re more than 2 minutes behind, don’t wait until another 5 questions have passed and you realize you’re 5 minutes behind. At this point, you want to find a question in the next 5 that you can guess randomly on. The quicker you can identify a good candidate to skip, the more time you can make up.

This is another scenario where random guessing is best. Educated guessing takes time, and we’re trying to save as much time as possible. Look for questions that take a long time to read, or that deal with topics you’re not as strong in, but most importantly, just make the decision and pick up the time.

Wrap Up

Remember, this test is not like high school exams; it’s not designed to have every question answered. This test is about consistency on questions you know how to do. Knowing when to get out of a question is one of the most fundamental parts of a good score. The better you are at limiting time spent on really difficult questions, the more time you have to answer questions you know how to do.

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AppIconWe are happy to announce that the latest version of our free GMAT app, Pocket GMAT Flashcards, is now available for download via the App store! New updates include:

  • Back-end and usability fixes
  • Content overhaul
  • Updated for iOS7
  • Shiny new icon

Containing over 350 GMAT quant flash cards, Pocket GMAT uses an adaptive algorithm developed by Manhattan Prep instructors to help you target cards you most need help with. Allowing you to strengthen your GMAT quantitative skills anywhere and at any time, the Pocket GMAT app is an indispensable tool for iPhone users.

The app also now works better on iOS6 devices and we have fixed issues with scrolling and swiping, so overall navigation is smoother. We’ve also fixed content errata and made the images look better.

Manhattan Prep has teamed up with Learningpod to make Pocket GMAT free for everyone! In addition to the adaptive algorithm, there is also a sequential practice mode that lets you flip through the cards however you want. You also have the ability to enter a Target Date to keep you on pace and track your progress. The flash cards are organized into “KeyRings” by topic and include algebra, number properties, word problems, geometry, fractions, decimals, and percents.

We hope the new updates improve your studying experience, and if you’re as excited as we are about the revisions, please let us know in the review section of the App store. We use your feedback to make our study tools the best they can possibly be!

gmat-quantStop! Before you dive in and start calculating on a math problem, reflect for a moment. How can you set up the work to minimize the number of annoying calculations?

Try the below Percent problem from the free question set that comes with your GMATPrep® software. The problem itself isn’t super hard but the calculations can become time-consuming. If you find the problem easy, don’t dismiss it. Instead, ask yourself: how can you get to the answer with an absolute minimum of annoying calculations?

 

District

Number of Votes

Percent of Votes for Candidate P

Percent of Votes for Candidate Q

1

800

60

40

2

1,000

50

50

3

1,500

50

50

4

1,800

40

60

5

1,200

30

70

 

* ” The table above shows the results of a recent school board election in which the candidate with the higher total number of votes from the five districts was declared the winner. Which district had the greatest number of votes for the winner?

“(A) 1

“(B) 2

“(C) 3

“(D) 4

“(E) 5”

 

Ugh. We have to figure out what they’re talking about in the first place!

The first sentence of the problem describes the table. It shows 5 different districts with a number of votes, a percentage of votes for one candidate and a percentage of votes for a different candidate.

Hmm. So there were two candidates, P and Q, and the one who won the election received the most votes overall. The problem doesn’t say who that was. I could calculate that from the given data, but I’m not going to do so now! I’m only going to do that if I have to.

Let’s see. The problem then asks which district had the greatest number of votes for the winner. Ugh. I am going to have to figure out whether P or Q won. Let your annoyance guide you: is there a way to tell who won without actually calculating all the votes?

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GMAT-geometryA couple of months ago, we talked about what to do when a geometry problem pops up on the screen. Do you remember the basic steps? Try to implement them on the below GMATPrep® problem from the free tests.

* ”In the xy-plane, what is the y-intercept of line L?

“(1) The slope of line L is 3 times its y-intercept
“(2) The x-intercept of line L is – 1/3”

My title (3 Steps to Better Geometry) is doing double-duty. First, here’s the general 3-step process for any quant problem, geometry included:

Screen Shot 2014-02-05 at 12.13.43 PM

All geometry problems also have three standard strategies that fit into that process.

First, pick up your pen and start drawing! If they give you a diagram, redraw it on your scrap paper. If they don’t (as in the above problem), draw yourself a diagram anyway. This is part of your Glance-Read-Jot step.

Second, identify the “wanted” element and mark this element on your diagram. You’ll do this as part of the Glance-Read-Jot step, but do it last so that it leads you into the Reflect-Organize stage. Where am I trying to go? How can I get there?

Third, start Working! Infer from the given information. Geometry on the GMAT can be a bit like the proofs that we learned to do in high school. You’re given a couple of pieces of info to start and you have to figure out the 4 or 5 steps that will get you over to the answer, or what you’re trying to “prove.”

Let’s dive into this problem. They’re talking about a coordinate plane, so you know the first step: draw a coordinate plane on your scrap paper. The question indicates that there’s a line L, but you don’t know anything else about it, so you can’t actually draw it. You do know, though, that they want to know the y-intercept. What does that mean?

They want to know where line L crosses the y-axis. What are the possibilities?

Infinite, really. The line could slant up or down or it could be horizontal. In any of those cases, it could cross anywhere. In fact, the line could even be vertical, in which case it would either be right on the y-axis or it wouldn’t cross the y-axis at all. Hmm.
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challenge problem
We invite you to test your GMAT knowledge for a chance to win! The second week of every month, we will post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that month’s drawing for free Manhattan GMAT prep materials. Tell your friends to get out their scrap paper and start solving!

Here is this month’s problem:

If pq, and r are different positive integers such that p + q + r = 6, what is the value of x ?

(1) The average of xp and xq is xr.

(2) The average of xp and xr is not xq.

Continue Reading…